1. Technical Field
The present invention relates generally to optical lithography, and more particularly to calculating image intensity of a mask by decomposing a Manhattan polygon based on parallel edges.
2. Background Art
The design of Very-Large Scale Integrated (VLSI) or Ultra-Large Scale Integrated (ULSI) circuits requires analysis of the images produced by projection lithography systems. Due to the scale of this analysis, large computational resources are required. Using the information specific to the object (i.e., the mask) in a lithographic system, techniques have been developed to quickly generate the resulting intensity at a location of interest on the imaging plane. These techniques are widely utilized in model-based optical proximity correction (OPC) to determine the appropriate compensating adjustments to the integrated circuit (IC) shapes deployed on the mask. The mask can, in almost all cases, be represented as a number of two-dimensional (2D) polygon apertures. Such 2D polygons neglect the finite thickness of the mask film structure, and are thus known as a thin-mask representation. When the true finite-thickness topography is included, the mask is referred to as a topographic mask. A fast method for generating the impact at an image point from a polygon is derived from the Hopkins model. Two techniques are usually combined together to achieve a high-speed simulation module: Sum Of Coherent Systems (SOCS) decomposition and table lookup for convolution. In the SOCS decomposition, partial coherent imaging under the Hopkins model is approximately decomposed into an incoherent sum of a relatively small number of coherent sub-images. Here the term “sub-image” refers to the convolution of an eigenvector kernel of the Hopkins Integral (used in the Hopkins model) with the object, an operation equivalent to calculating a coherent image. The full image is obtained as the incoherent sum of the sub-images. Such an eigenvector kernel is referred to as a SOCS kernel. A table lookup will reduce the computational expense to obtain the coherent sub-image since each mask polygon can be decomposed into a superposition of unbounded sectors, and the coherent sub-images of each kind of unbounded sector are pre-computed and stored in a table.
In most cases the polygons are Manhattan polygons, whose edges are rectilinear with respect to the boundaries of the rectangular IC chip, with directions being referred to as either horizontal or vertical. Only one type of unbounded sector, i.e., a quarter-plane element, is needed to synthesize the internal area of a Manhattan polygon. Where structures are periodic (or assumed periodic for simulation purposes), the quarter-plane element needs not actually extend indefinitely to occupy an entire quadrant of the unbounded 2D mask plane, but may instead be terminated at the period boundaries. For simplicity the term “quarter-plane” will be used to refer to such elements used under either option.
Conventionally, as shown in FIG. 1, an arbitrary polygonal area 10 of Manhattan form (shown as shaded area) in the left part of FIG. 1 can be formed by superposing (or be decomposed into) quarter-planes 12a-12h (each may be generally referred to as quarter-plane 12) defined by each corner 11a-11h (each may be generally referred to as a corner 11) of polygon 10, respectively. Note that quarter-planes 12 must be superposed with alternately positive and negative sign (as shown by the “+” and “−” references in FIG. 1) to indicate whether a quarter-plane is added or subtracted from the mask plane. When quarter-planes 12 are added or subtracted from the mask plane in this way, the net residual added area is positive, and takes the form of the desired polygonal area 10.
In the conventional table lookup scheme, the lookup tables store the coherent sub-images of quarter-plane elements addressed by only a single coordinate point, e.g., the lower left corner (as shown in FIG. 1) or upper left corner of quarter-plane elements 12, as convolved with kernels from a Hopkins Matrix (or integral), and is called a corner lookup scheme. The way in which quarter-planes are used for the convolution is based on the decomposition method for Manhattan polygons described above with respect to Manhattan polygon 10 of FIG. 1. Without loss of generality, a quarter-plane 12's lower left corner is defined by a corner 11 of Manhattan polygon 10. Because of the linearity of SOCS approximation, the convolution between a Manhattan polygon and a kernel can be decomposed to the convolutions between quarter-plane elements 12 addressed by corners 11 of polygon 10 and the kernel, and all these convolutions that might be encountered will be pre-computed and stored in a lookup table(s).
Access to the lookup table during OPC involves a more-or-less random selection of entries, and in general random access to large data tables entails delays that are quite long on the time-scale of central processing unit (CPU) operations. As a result, the speed of today's OPC technology is strongly gated by memory access delays. OPC processing on each single chip layer currently requires many days or even weeks of processing on very large state-of-the-art computer clusters, which limits the speed at which semiconductor products can be produced.
As mask features get smaller and smaller, electromagnetic effects (EMF) will have a more and more substantial impact on the image calculation. When mask features are large compared to the wavelength, only consideration of the interior areas of the mask features is required when calculating the image, and the finite thickness of the mask films (which manifests itself as a topographic discontinuity at feature edges) can be ignored. This simple model maintains some accuracy even at the fine dimensions of today's IC features because lithographic masks are formed at an enlarged scale (typically 4× enlarged) and their polygonal features are demagnified by several folds when projected onto the wafer to form the circuit features. Even so, state-of-the-art simulations for OPC require that the EMF effects be accounted for, at least approximately. In the prior-art, the complex electromagnetic iteration between the incident illumination wave and the mask topography has been shown to be highly localized to the vicinity of the mask topography edges. Based on this observation, imaging with EMF considered is carried out by adding localized adjustments to the edges of the polygons. Examples of such adjustments include the Boundary Layer method (BL) and the edge-based Domain Decomposition Method (DDM). Until recently, the convolution of such an adjustment with each SOCS kernel requires an additional table lookup of a separate table. In addition, in the usual case of unpolarized illumination, it was necessary to calculate separate images for each independent polarization component of the illumination, and then combine them to form the complete image.